 GroupTheory - Maple Programming Help

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GroupTheory

 DihedralGroup
 construct a dihedral group of a given degree

 Calling Sequence DihedralGroup( n ) DihedralGroup( n, s )

Parameters

 n - a positive integer s - (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)

Description

 • The dihedral group of degree $n$ is the symmetry group of an $n$-sided regular polygon for $n>2$. It is generated by a reflection (of order $2$), and a rotation (of order $n$). It acts as a permutation group on the vertices of the regular $n$-sided polygon.
 • For $n=1$, the dihedral group is a cyclic group of order $2$.  For $n=2$, the dihedral group is the non-cyclic group of order $4$, also known as the Klein $4$-group.
 • The DihedralGroup( n ) command returns a dihedral group, either as a permutation group or a group defined by generators and defining relations. By default, a permutation group is returned, but a finitely presented group can be requested by passing the option 'form' = "fpgroup".
 • If the value of the parameter n is not numeric, then a symbolic group representing the dihedral group of the indicated degree is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{DihedralGroup}\left(13\right)$
 ${{\mathrm{D}}}_{{13}}$ (1)
 > $\mathrm{DihedralGroup}\left(13,\mathrm{form}="fpgroup"\right)$
 ${{\mathrm{D}}}_{{13}}$ (2)
 > $\mathrm{DihedralGroup}\left(13,\mathrm{form}="permgroup"\right)$
 ${{\mathrm{D}}}_{{13}}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{DihedralGroup}\left(3k\right)\right)$
 ${6}{}{k}$ (4)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(6k\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::'\mathrm{posint}'$
 ${\mathrm{false}}$ (5)

Compatibility

 • The GroupTheory[DihedralGroup] command was introduced in Maple 17.